where \(\eta ^\pm _X\)  is the inclusion of the attracting (resp. repelling) subset, and \(p^+_X\)  (resp. \(p^-_X\) ) sends \(x\)  to \(\lim _{t\rightarrow 0}x\)  (resp. \(\lim _{t\rightarrow \infty }x\) ). This diagram is easy to describe when \(X\)  is an affine scheme: therefore it suffices to find \(\mathbb {C}^\ast \) -equivariant covering of \(X\)  by affine spaces. The existence of such covering is not obvious at all, and is far too restrictive in the Zariski topology. Fortunately, a deep result of [1]} and [2]} shows that quasi-separated algebraic spaces locally of finite type have a \(\mathbb {C}^\ast \) -equivariant étale cover by affine spaces. For an algebraic stack \(\mathcal {X}\)  with a \(\mathbb {C}^\ast \) -action, consider \(\mathcal {Y}:=[\mathcal {X}/T]\) ; for \(x\in \mathcal {X}\) , considering the corresponding point \(y\in \mathcal {Y}\) , there is an exact sequence:
\(\begin{tikzcd}1[r] & G_x[r] & G_y[r] & T_x[r] & 1\end{tikzcd}\) 
